Dual Wielder only allows drawing/stowing a second weapon
Your plan would fall over at the last step because you only get one free object interaction per turn.
From "Other Activity on Your Turn" (PHB, p. 190):
You can also interact with one object or feature from the environment for free, during either your move or your action. For example, ... you could draw your weapon as part of the same action you use to attack.
If you want to interact with a second object, you need to use your action.
The Dual Wielder feat does not give you two free item interactions, it lets you draw or stow two weapons at the same time instead of one. Drawing and stowing the same weapon clearly cannot be done simultaneously, so this would require two item interactions to achieve.
The relevant benefit of the Dual Wielder feat (PHB, p. 165):
- You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.
As it stands, your plan would only work if you waited until you next turn to draw the whip again or if you found a way to have two actions (e.g. haste, Action Surge, etc.) so that you have an extra action with which to draw the whip again.
That said, green-flame blade in particular, as I understand it, wouldn't require you to put the whip away, since it doesn't require a Somatic component and the Material component is "a weapon" (SCAG, p. 143). It goes on to say:
As part of the action used to cast this spell, you must make a melee attack with a weapon...
So you could skip all this whip juggling and just cast the spell with your action, using your blade in hand (the other hand being irrelevant).